New interpretation of complex hyperbolic form as Weil-Petersson form.
problem Understanding complex hyperbolic structures on moduli spaces.
method Interpreting complex hyperbolic form as a Weil-Petersson form for punctured spheres.
result Found a new equality between two symplectic forms.
Hyperbolic solitons on trans-Sasakian space forms and their submanifolds
problem Characterizing hyperbolic solitons on trans-Sasakian space forms and their submanifolds
method Introducing hyperbolic ∗−Ricci solitons and hyperbolic Ricci-Yamabe solitons result Characterizing the nature of hyperbolic solitons on trans-Sasakian space forms and their submanifolds
The study examines curvature tensors and solitons in Lorentzian trans-Sasakian space forms.
problem Characterizing curvature tensors and solitons in Lorentzian trans-Sasakian space forms.
method Derivation of various curvature tensors and analysis of solitons under specific conditions.
result Conditions for hyperbolic Ricci and conformal Ricci solitons to be η-Einstein and their expansion/steering/shrinking properties. The paper studies the Poisson transform of differential forms on hyperbolic spaces.
problem Analyzing the Poisson transform of differential forms on real hyperbolic spaces.
method Proving the Poisson transform is a topological isomorphism between boundary forms and eigenforms.
result The Poisson transform is a topological isomorphism for Lr-differential forms on the boundary of hyperbolic spaces. Minimum algebraic intersection found in hyperbolic surfaces, growing with genus.
problem Finding the minimum algebraic intersection form in hyperbolic surfaces.
method Analyzing algebraic intersection form in moduli space of hyperbolic surfaces.
result Minimum grows in the order of (logg)−2 with genus. The paper explores properties of projections and gradient methods in hyperbolic space forms.
problem Optimization problems in hyperbolic space forms.
method Intrinsic κ-projection and gradient projection methods.
result Every accumulation point of the sequence generated by the gradient projection method is a stationary point.
Infinite volume moduli spaces of hyperbolic surfaces are redefined with exponential forms.
problem Infinite volume of moduli spaces of hyperbolic surfaces with cusps.
method Introduce exponential volume form exp(-W)Vol(K,L) where W is a function of hyperbolic areas.
result Exponential volume forms make moduli spaces finite and relevant to open string theory.
In this paper, we prove the nonexistence of L2 harmonic 1-forms on a complete super stable minimal submanifold M in hyperbolic space under the assumption that the first eigenvalue λ1(M) for the Laplace operator on M is bounded below by (2n−1)(n−1). Moreover, we provide sufficient conditions for minimal sub…
The paper studies rigidity and stability of minimal submanifolds in hyperbolic space.
problem Conditions for a minimal submanifold to be totally geodesic.
method Analyzes the length of the second fundamental form and eigenvalues of the super stability operator.
result Sharp upper bounds for the first eigenvalue of the super stability operator for surfaces in hyperbolic 4-space.
Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths are sent to infinity, the Weil-Petersson form converges to a piecewise linear form first defined by Kontsevich. The proof …
The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form. It is shown that several conditions on the Gaussian curvature of the second fundamental form can be satisfied only by extrinsic spheres.
The Euler number of special symplectic hyperbolic manifolds is positive.
problem Understanding the Euler number of symplectic hyperbolic manifolds.
method Study L2-harmonic forms on the universal covering space and prove the Singer conjecture. result The Euler number of a special symplectic manifold satisfies (−1)nχ(X)>0. The space of broken hyperbolic structures generalizes the Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations (equivalently, the space of projectivized affine foliations) generalizes the space of projectivized measured foliations. Just as projectivized measured foliations…
In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a pinching condition in a hyperbolic space form to a round point in finite time.
This paper connects spinors to horospheres in hyperbolic space.
problem Understanding geometric relationships between spinors and horospheres in hyperbolic space.
method Explicit bijective correspondence between spinors and horospheres, using bilinear forms and complex-valued distances.
result Derived applications include Ptolemy equations and Plücker coordinates connections.
Unique inhomogeneous ruled hypersurface found in complex hyperbolic space.
problem Classifying ruled real hypersurfaces with constant norm.
method Analyzing nonflat complex space forms, proving existence and uniqueness.
result Existence of a unique inhomogeneous example in complex hyperbolic space.
Study vector fields on hyperbolic spaces to create Ricci-Bourguignon solitons.
problem Characterize vector fields on hyperbolic spaces Hn that transform them into Ricci-Bourguignon solitons. method Detailed geometric study of vector fields in dimensions n=2,3 and n≥3, focusing on dual forms in odd dimensions. result Dual forms of these vectors are contact forms in odd dimensions.
Study immersions of surfaces into SL(2,C) and geodesics space.
problem Classical theory of hypersurfaces in pseudo-Riemannian space forms.
method Holomorphic Riemannian space forms and geodesics space of hyperbolic space.
result Characterization and construction of immersions.
The space forms, the complex hyperbolic spaces and the quaternionic hyperbolic spaces are characterized as the harmonic manifolds with specific radial eigenfunctions of the Laplacian.
We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this theorem, the first one is to show an analogue of the Ahlfors islands theorem for it and the second one is to give a simple proof of t…
Study examines Kähler immersions of ALE Kähler metrics into complex space forms.
problem Kähler immersions of ALE Kähler metrics into complex space forms.
method Investigation of the relationship between Kähler immersions and the mass of ALE Kähler metrics.
result ALE Kähler metrics with positive mass do not admit a Kähler immersion into complex Euclidean space.
We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are hyperbolic space forms.
We describe smooth compactifications of certain families of reductive homogeneous spaces such as group manifolds for classical Lie groups, or pseudo-Riemannian analogues of real hyperbolic spaces and their complex and quaternionic counterparts. We deduce compactifications of Clifford-Klein forms of these homogeneous sp…
The paper proves inequalities for convex hypersurfaces in spheres and hyperbolic spaces.
problem Understanding geometric properties of convex hypersurfaces in curved spaces.
method Proving identities and inequalities for closed, strictly convex hypersurfaces in spheres and hyperbolic/de Sitter space.
result Generalized Blaschke-Santaló type inequalities and quermassintegral inequalities in hyperbolic/de Sitter space.
In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface M in the hyperbolic space which has finite L2-norm of the second fundamental form on M. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stabl…
Study finds optimal loops in hyperbolic space with Finsler structure.
problem Optimal loops in Finsler hyperbolic plane.
method Left-invariant Finsler structure, convex trigonometry functions.
result Optimal isoperimetric loops found in terms of trigonometry functions.
The paper examines ancient solutions of mean curvature flow in space forms with curvature pinching conditions.
problem Investigating rigidity of ancient solutions of mean curvature flow in space forms.
method Sharp asymptotic pointwise curvature pinching conditions and asymptotic integral curvature pinching conditions.
result Ancient solutions in a sphere are either a shrinking spherical cap or a totally geodesic sphere, and in a hyperbolic space, they are a family of shrinking spheres.
The paper characterizes biconservative surfaces in hyperbolic 4-space.
problem Characterizing biconservative surfaces in hyperbolic 4-space.
method Establishing intrinsic conditions and analyzing geometric properties.
result Biconservative surfaces in hyperbolic 4-space satisfy a specific intrinsic condition.
Study topological hyperbolicity of moduli spaces of elliptic surfaces.
problem Characterize the largeness of the topological fundamental group of complex varieties.
method Introduce topological hyperbolicity and provide supporting evidence for moduli spaces of elliptic surfaces.
result Establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one.
Examines discrete curvature's relation to smooth curvature in 3 spaces.
problem Understanding how discrete curvature relates to smooth curvature in different spaces.
method Using specific triangular tilings of 3 types of spaces to examine curvatures.
result Discrete curvature can sense the smooth curvature of ambient space forms.
Classifies actions on complex space forms with Lagrangian orbits.
problem Classifying actions on complex space forms with Lagrangian orbits.
method Classifies holomorphic isometric actions on complex space forms.
result Only examples are Lagrangian affine subspace foliations of complex Euclidean spaces and Lagrangian horocycle foliations of complex hyperbolic spaces.
The paper proves new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
problem Proving new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
method Locally constrained inverse curvature flows in hyperbolic and spherical spaces.
result Established new Alexandrov-Fenchel and Minkowski inequalities involving general convex weight functions.
Study on a specific type of Riemannian manifolds constructed from 2D space-forms.
problem Characterizing and understanding new types of Riemannian manifolds.
method Constructed as a product of a real line and a 2-dimensional Riemannian space-form, with metrics derived from cone and hyperbolic extensions.
result Characterized and studied in terms of their curvature properties.
Using hyperbolic form convolution with doubly isometry-invariant kernels, the explicit expression of the inverse of the de Rham laplacian acting on m-forms in the Poincaré space is found. Also, by means of some estimates for hyperbolic singular integrals, we obtain L^p-estimates for the Riesz transforms passing from th…
Study submanifolds with relative nullity in space forms using splitting tensor.
problem Characterize submanifolds with relative nullity in space forms.
method Use splitting tensor and Codazzi equation to express second fundamental form.
result Derive new strong consequences in hyperbolic and Euclidean spaces.
A new method defines bounded cohomology classes from differential forms.
problem Defining bounded cohomology classes from differential forms.
method Integration over geodesic simplices and Fourier analysis on the hyperbolic plane.
result The method defines an injective embedding of differential forms into bounded cohomology classes.
The paper studies timelike minimal Lagrangian surfaces in indefinite complex hyperbolic space.
problem Characterizing timelike minimal Lagrangian surfaces in indefinite complex hyperbolic space.
method Defining natural Gauss maps and proving a Ruh-Vilms type theorem.
result Timelike minimal Lagrangian surfaces correspond to the fifth real form of the complex affine Kac-Moody algebra.
Convexity properties of Weil-Petersson geodesics on the Teichmüller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considera…
We generalize Conway's approach to integral binary quadratic forms on Q to study integral binary hermitian forms on quadratic imaginary extensions of Q. In Conway's case, an indefinite form that doesn't represent 0 determines a line ("river") in the spine T associated with SL(2,Z) in the hyperbolic plane. In our genera…
We show the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces on differential forms, including high energy estimates in strips. This is achieved by placing the spectral family of the Laplacian within the framework developed, and applied to scalar problems, by the author recentl…
Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.
We show that an isoparametric submanifold of a complex hyperbolic plane, according to the definition of Heintze, Liu and Olmos', is an open part of a principal orbit of a polar action. We also show that there exists a non-isoparametric submanifold of the complex hyperbolic plane that is isoparametric according to the d…
Study linking numbers in hyperbolic 3-folds, linking to Siegel modular forms.
problem Linking numbers between geodesics in arithmetic hyperbolic 3-folds.
method Analyzing integral of Kudla--Millson theta series over Seifert surfaces.
result Series converges to genus 2 Siegel modular forms of weight 2.
The paper finds the maximum scalar curvature for 3D minimal hypersurfaces in hyperbolic 4-space.
problem Finding the maximum scalar curvature for 3D minimal hypersurfaces in hyperbolic 4-space.
method Using the Generalized Maximum Principle, the paper proves that a 3D complete minimal hypersurface with constant scalar curvature in H4(−1) satisfies S≤2921. result A 3D complete minimal hypersurface in H4(−1) with constant scalar curvature satisfies S≤2921. In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the …
We prove that if an n-dimensional complete minimal submanifold M in hyperbolic space has sufficiently small total scalar curvature then M has only one end. We also prove that for such M there exist no nontrivial L2 harmonic 1-forms on M.
We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.
New inequalities for convex hypersurfaces in various spaces.
problem Deriving inequalities for hypersurfaces under convex weight.
method Sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces in Euclidean, spherical, and hyperbolic spaces.
result Incorporates convex, non-decreasing positive functions as weights, yielding a broad family of geometric inequalities.